Thread: Continuous Functions!

Suppose h : (0,1)-> satisfies the following conditions:
for all x Э (0,1) there exists d>0 s.t. for all x' Э (x, x+d)n(0,1) we have h(x)<=h(x')

Prove that if h is continuous on (0,1) then h(x)<=h(y) whenever x,y Э (0,1) and x<=y. Use a counterexample to show that this results may not be true when h is continuous

I thought I could do this by using definitions, but its not working for me.

Originally Posted by emjones

Suppose h : (0,1)-> satisfies the following conditions:
for all x Э (0,1) there exists d>0 s.t. for all x' Э (x, x+d)n(0,1) we have h(x)<=h(x')

Prove that if h is continuous on (0,1) then h(x)<=h(y) whenever x,y Э (0,1) and x<=y. Use a counterexample to show that this results may not be true when h is continuous

I thought I could do this by using definitions, but its not working for me.

I hate reading plain text

Suppose 0,1)\to\mathbb{R}" alt="h0,1)\to\mathbb{R}" /> (presumably) satisfies the following conditions: for every there exists some such that . Prove that if is continuous then is non-decreasing. Give a counter example to show the reverse is isn
t true.

Right? What have you tried?

Hi drexel, sorry I don't know how to use the codes, but thank you for changing it for me.

I have tried to use a corollary in my notes relating to monotonic functions and differentiability, where the MVT is applied to [x,y]<[a,b] but im not getting anywhere.

Also I was asked to state the IVT earlier in the question, so thought this might be of use, but I don't think it can be used earlier! This is why im confused! Please help!

Originally Posted by emjones

Hi drexel, sorry I don't know how to use the codes, but thank you for changing it for me.

I have tried to use a corollary in my notes relating to monotonic functions and differentiability, where the MVT is applied to [x,y]<[a,b] but im not getting anywhere.

Also I was asked to state the IVT earlier in the question, so thought this might be of use, but I don't think it can be used earlier! This is why im confused! Please help!

You can't use the MVT! How do you know that is differentiable? How do you think you'd use the IVT, that's what I'm thinking too.